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hi everyone today we're going to talk about how to use transformations to sketch the graph
of a function to complete this problem we'll first sketch a small piece of our function
then modify it one transformation at a time until it represents the complete function
let's take a look in this particular problem we've been asked to use our knowledge of transformations
to sketch the graph of y equals one half e to the negative x minus one and all I want
to show you here is that you can sketch the graph of a function that you know and then
you use your knowledge of transformations to build out the rest of your function so
what I mean by that is we could for example recognize that we have e to the x this function
here or this part of our function e to negative x is just e to the x but with a negative x
substituted in so what we could do is start with the graph of e to the x if we know what
that looks like it's a common function that might be a graph that we know we can start
with that and then we can change e to the x into e to the negative x and then we can
apply the one half and then apply the negative one so here's what that could look like we
could sketch the graph of e to the x and that looks something roughly like this you may
know that it's a curve that looks like this it intersects the y axis at y equals one and
this would be e to the x but we don't have e to the x we have e to the negative x so
what we want to do is recognize that all we've done here right is this function is f of x
equals e to the x well we've gone ahead and said f of negative x because we're plugging
in negative x for x well what we should know from our knowledge of transformations or study
of transformations is that if you plug in negative x for x into a function it's just
going to flip the graph across the y axis so we're going to have the same function except
that it's going to be backwards like this flipped across the y axis so again we'll intersect
the graph at intersect the y axis at y equals one and the graph is just flipped across the
y axis like this so now we have the graph of e to the negative x well our next step
is to apply this one half here right we need one half times e to the negative x so what
we want to do in that case is say this is going to be the graph of f of x equals one
half e to the negative x and we know from transformations that multiplying by one half
means that we're going to shrink the graph vertically by a factor of two so instead of
the graph intersecting the y axis at one it needs to intersect at one half and every y
coordinate will be half what it is so if this y coordinate here for example were four our
new y coordinate here will be two y equals two so we just shrink by a factor of two so
I may not draw this perfectly but maybe what that looks like would be something like this
we intersect at one half we know that and we would have every y coordinate would be
half of what it use to be and that would be one half e to the negative x and then our
final step would be to apply this negative one here we know from studying transformations
that adding a negative one like that is just going to shift our graph down one unit so
if we're talking about f of x equals one half e to the negative x minus one we have the
same graph that we did over here except we move it down one unit so instead of intersecting
the y axis here at one half we're going to be intersecting at negative one half so that
will look something roughly like call this negative one half here something roughly like
this maybe negative one half here and that would be the curve like that so it's a little
rough but you get the idea so as you can see we started with e to the x and we just manipulated
that one step at a time using what we already know about transformations to sketch the final
graph of y equals one half e to the negative x minus one so I hope you found that video
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